f(x) = 3x^4 + 17x^3 + 9x^2 – 7x – 10; g(x) = x + 5

Question

Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not:

f(x) = 3x⁴ + 17x³ + 9x² – 7x – 10; g(x) = x + 5

Answer 1

If g(x) is a factor of f(x), then the remainder will be zero that is g(x) = 0. 

g(x) = x + 5 = 0, then x = -5

Remainder = f(-5)

Now, f(3) = 3(-5)⁴ + 17(-5)³ + 9(-5)² – 7(-5) – 10

= 3 x 625 + 17 x (-125) + 9 x (25) – 7 x (-5) – 10

= 1875 - 2125 + 225 + 35 – 10 = 0

Therefore, g(x) is a factor of f(x).